List of ordinals

Countables

• 0, the smallest ordinal
• 1, the first successor ordinal
• \( \omega \), the first limit ordinal
• \( \omega\cdot2 \)
• \( \omega^{2} \)
• \( \omega^{3} \)
• \( \omega^{\omega} \)
• \( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)
• \( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)
• \( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)
• \( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \), the Feferman-Schutte ordinal and the PTO of \( \text{ATR}_{0} \)
• \( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \), the Ackermann Ordinal
• \( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \), the SVO (Small Veblen ordinal)
• \( \psi_{0}(\Omega^{\Omega^{\Omega}}) \), the LVO (Large Veblen ordinal)
• \( \psi_{0}(\Omega_{2}) \), the BHO (Bachmann-Howard ordinal)
• \( \psi_{0}(\Omega_{\omega}) \), the BO (Buchholz ordinal)
• \( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \), the TFB (Takeuti-Feferman-Buchholz ordinal)
• \( \psi_{0}(\Omega_{\Omega_{\dots}}) \), the EBO (Extended Buchholz ordinal)
• \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of \( \text{KPi} \)
• \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \text{KPM} \)
• \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \text{KP} + \Pi_{3}\text{-refl.} \)
• \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \text{KP} \) with a \( \Pi_{\mathbb{N}}\text{-refl.} \) universe under ZF + V = L
• \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second OCF using indescribable cardinals
• PTO of \( \text{Z}_{2} \)
• PTO of \( \text{ZFC} \)
• \( \omega^{\text{CK}}_{1} \)
• RECURSIVE ORDINALS GO HERE
• STABILITY STUFF GOES HERE
• Infinite time Turing machine ordinals
  • \( \lambda \), the supremum of all writable ordinals
  • \( \gamma \), the supremum of all clockable ordinals
  • \( \zeta \), the supremum of all eventually writable ordinals
  • \( \Sigma \), the supremum of all accidentally writable ordinals
• The smallest gap ordinal

Uncountables

• \( \Omega \), the smallest uncountable ordinal
• \( I \), the smallest inaccessible ordinal
• \( M \), the smallest mahlo cardinal
• \( K \), the smallest weakly compact cardinal